Theorem ifpr 2398

Description: Membership of a conditional operator in an unordered pair.

Assertion

Ref	Expression
ifpr	|- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})

Proof of Theorem ifpr

Step	Hyp	Ref	Expression
1		ifcl 2351	. . 3 |- ((A e. V /\ B e. V) -> if(ph, A, B) e. V)
2		ifor 2352	. . . 4 |- (if(ph, A, B) = A \/ if(ph, A, B) = B)
3		elprg 2394	. . . 4 |- (if(ph, A, B) e. V -> (if(ph, A, B) e. {A, B} <-> (if(ph, A, B) = A \/ if(ph, A, B) = B)))
4	2, 3	mpbiri 194	. . 3 |- (if(ph, A, B) e. V -> if(ph, A, B) e. {A, B})
5	1, 4	syl 10	. 2 |- ((A e. V /\ B e. V) -> if(ph, A, B) e. {A, B})
6		elisset 1792	. 2 |- (A e. C -> A e. V)
7		elisset 1792	. 2 |- (B e. D -> B e. V)
8	5, 6, 7	syl2an 454	1 |- ((A e. C /\ B e. D) -> if(ph, A, B) e. {A, B})

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 1099   e. wcel 1105  Vcvv 1786  ifcif 2332  {cpr 2381

This theorem is referenced by:  suppr 4514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-un 2021  df-if 2333  df-sn 2383  df-pr 2384

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